 Ordered Čech cochain complex

For a well-order on the indices of an open cover, the ordered Čech cochain complex uses only strictly increasing tuples, with the alternating restriction Čech differential. It computes the same Čech cohomology as the complex using all tuples. If every finite intersection is nonempty and the coefficient constant sheaf has the same abelian group $A$ on each intersection, inserting the least index $\alpha$ gives a cochain homotopy: set $(hs{)}_I=s_{\alpha ,I}$ when $\alpha$ is not in $I$, and $(hs{)}_I=0$ otherwise. Expanding the alternating sums gives $\delta h+h\delta =\mathrm{id}$ in every positive degree, since the terms deleting an index of $I$ cancel and the term deleting $\alpha$ is $s_I$. Hence positive Čech cohomology vanishes in this situation; degree zero remains $A$.